﻿Elastic Equilibrium under Surface Tractions, 503 



Both {<f> 1} <b 2 , <fc$, Xi> %2, Xs\ and {^1^2,^3,^1,^2,^3} 

 are to follow the latter type. 



The relations between these two sets in elastic equilibrium 

 must be linear. 



I shall take the invariant function (2V), which gives the 

 potential strain-function per unit of volume in the form, 



2V = \ n e 2 + 2\ 12 ef + 2\ l5 eg + 2\ u ea + 2\ 15 eb -r2\ u ec 

 + \ 22 f 2 + 2\ n fg +2\Ja + 2\ 25 fb 4- 2\ 26 /c 



+ ^33 9 2 + ^i9 a +2X 35 ^ + 2X 36 #c 



+ X 44 a 2 + 2\ i5 ab + 2X 46 ac 



+ X 55 6 2 + 2X 56 fo 



+ X 66 c 2 . 



... (3) 



Since V is to be an invariant function, that is, since 

 il 1 A" = 0, H 2 V = 0, I2 3 V = 0, we deduce that, acting on the 

 coefficients, 



0l = 2Xu {^u ~ 5^) +(Xl3 ~ Xl2) d^~ Xl6 ?5X7 5 +Xl5 ' 



dx 



16 



+ ^^^\ — h2(X 34 — ^24)^ h(2X44 — X 2 2+X 2 3)^— 



+ (2X 45 — X 26 ) ^r h (2X 46 + X 25 ) ^r— 



OA- 2 5 OA, 2 f? 



~" 4X 34 ^ (2X 44 — X 3 3+X 23 )^- (2X45+ X 36 )^— 



OX 33 OA34 0^35 



— (2X 46 — X 35 ). 



Bx 36 



+ 2(X 34 — X 24 )^- + (X 35 — X 25 — X 46 )^— - 



+ (^36 — ^26 + ^45)^ 



OA-46 



-2\ 56 J-+(X 55 -X 6(! ) S |-+2X 56S |- (4) 



From (3) we find the elastic values for P, Q, R, S, T, U 

 by differentiation in e, f, g, a, b, c respectively. Thus 



P = X n e + \ l2 f-\-\ n c/ + \ u a + \iob + \ l6 c, etc., 



and we may form an apparently suitable set of values for 



